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Tim S Evans

Imperial College London, Physics, Faculty Member

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I am a Senior Lecturer (equivalent to an associate professor) in the Theoretical Physics group and the Centre for Complexity Science at Imperial. My main interest is in the behaviour of many-body systems both in and out of equilibrium. Currently, I am interested in ideas falling under the broad area of complex systems. In particular the properties of Complex Networks, such as the "six degrees of separation", intrigue me. This is both from a theoretical perspective and in terms of applications to practical problems such as bibliometrics or cultural transmission, part of an interest in sociophysics in general. For instance, I have an ongoing project in Archaeology.More information is on my main site at Imperial College and on netplexity.org

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Papers by Tim S Evans

Are Copying and Innovation Enough?

Exact analytic solutions and various numerical results for the rewiring of bipartite networks are... more Exact analytic solutions and various numerical results for the rewiring of bipartite networks are discussed. An interpretation in terms of copying and innovation processes make this relevant in a wide variety of physical contexts. These include Urn models and Voter models, and our results are also relevant to some studies of Cultural Transmission, the Minority Game and some models of ecology. Comment: Contribution to the proceedings of ECMI08, based on a talk given by A.D.K.Plato as part of the minisymposium on Mathematics and Social Networks. Includes appendix of supplementary material not in published version

Network rewiring models

Networks & Heterogeneous Media, 2008

Recently we showed that a simple model of network rewiring could be solved exactly for any time a... more Recently we showed that a simple model of network rewiring could be solved exactly for any time and any parameter value. We also showed that this model can be recast in terms of several well known models of statistical physics such as Urn model and the Voter model. We also noted that it has been applied to a wide range of problems. Here we consider various generalisations of this model and include some new exact results.

Exact solution for the time evolution of network rewiring models

Physical Review E, 2007

We consider the rewiring of a bipartite graph using a mixture of random and preferential attachme... more We consider the rewiring of a bipartite graph using a mixture of random and preferential attachment. The full mean field equations for the degree distribution and its generating function are given. The exact solution of these equations for all finite parameter values at any time is found in terms of standard functions. It is demonstrated that these solutions are an excellent fit to numerical simulations of the model. We discuss the relationship between our model and several others in the literature including examples of Urn, Backgammon, and Balls-in-Boxes models, the Watts and Strogatz rewiring problem and some models of zero range processes. Our model is also equivalent to those used in various applications including cultural transmission, family name and gene frequencies, glasses, and wealth distributions. Finally some Voter models and an example of a Minority game also show features described by our model.

A Simple Model of Copying and Innovation

There are many situations where an 'individual' chooses an 'artifacts' by cop... more There are many situations where an 'individual' chooses an 'artifacts' by copying the existing choice of another individual. Names for new babies and registration rates of pedigree dogs often reflect current popular choices [1,2]. The allele for a particular gene carried ('chosen') by an individual reflects current gene frequencies [3]. In Urn models the probabilities controlling the urn chosen by a ball can reflect earlier choices [4]. The individuals in a Voter models [12] copy one of two choices made by a neighbour, as defined by a network of the individuals. Such copying is ...

Identifying time dependence in network growth

Physical Review Research, 2020

Linking the network centrality measures closeness and degree

Communications Physics

Measuring the importance of nodes in a network with a centrality measure is an core task in any n... more Measuring the importance of nodes in a network with a centrality measure is an core task in any network application. There many measures available and it is speculated that many encode similar information. We give an explicit non-linear relationship between two of the most popular measures of node centrality: degree and closeness. Based on a shortest-path tree approximation, we give an analytic derivation that shows the inverse of closeness is linearly dependent on the logarithm of degree. We show that our hypothesis works well for a range of networks produced from stochastic network models and for networks derived from 130 real-world data sets. We connect our results with previous results for other network distance scales such as average distance. Our results imply that measuring closeness is broadly redundant unless our relationship is used to remove the dependence on degree from closeness. The success of our relationship suggests that most networks can be approximated by shortest...

Cycle analysis of Directed Acyclic Graphs

Physica A: Statistical Mechanics and its Applications, 2022

Is network growth time dependent?

Identifying power-law scaling in real networks - indicative of preferential attachment - has prov... more Identifying power-law scaling in real networks - indicative of preferential attachment - has proved controversial. Critics argue that measuring the temporal evolution of a network directly is better than measuring the degree distribution when looking for preferential attachment. However, many of the established methods do not account for any potential time-dependence in the attachment kernels of growing networks, or methods assume that node degree is the key observable determining network evolution. In this paper, we argue that these assumptions may lead to misleading conclusions about the evolution of growing networks. We illustrate this by introducing a simple adaptation of the Barabási-Albert model, the "k2 model", where new nodes attach to nodes in the existing network in proportion to the number of nodes one or two steps from the target node. The k2 model results in time dependent degree distributions and attachment kernels, despite initially appearing to grow as line...

Modelling Cultural Dynamics: A Macroscopic Approach to Cultural Transmission in the Prehistoric Aegean*

INTRODUCTION In this paper we pursue a novel transdisciplinary approach to the question of cultur... more INTRODUCTION In this paper we pursue a novel transdisciplinary approach to the question of cultural dynamics, focussing particularly on the transmission of cultural traits across sociophysical space. In order to think through some of the themes involved we take as a case study an inter-regional phenomenon known as ‘Minoanisation’ (Wiener 1990; Broodbank 2004). This term describes a set of processes observed in the prehistoric Aegean (specifically the Middle and early Late Bronze Age) whereby ‘Minoan’ cultural traits find themselves transmitted beyond the Minoan sphere, i.e. the island of Crete. That is to say, communities in regions such as the Cyclades, the Dodecanese, coastal Asia Minor and the Greek mainland adopt aspects of Cretan culture, both in the form of actual imports from Crete and also local imitations thereof. These cultural elements include pottery shapes and styles, stone vases, loomweights and wall paintings; not only new forms of material culture, but also, presumab...

Time and Citation Networks

ArXiv, 2015

Citation networks emerge from a number of different social systems, such as academia (from publis... more Citation networks emerge from a number of different social systems, such as academia (from published papers), business (through patents) and law (through legal judgements). A citation represents a transfer of information, and so studying the structure of the citation network will help us understand how knowledge is passed on. What distinguishes citation networks from other networks is time; documents can only cite older documents. We propose that existing network measures do not take account of the strong constraint imposed by time. We will illustrate our approach with two types of causally aware analysis. We apply our methods to the citation networks formed by academic papers on the arXiv, to US patents and to US Supreme Court judgements. We show that our tools can reveal that citation networks which appear to have very similar structure by standard network measures turn out to have significantly different properties. We interpret our results as indicating that many papers in a bib...

Articulating scales in sea-based social Networks: A transdisciplinary approach

Dynamic networks and Hamiltonian landscapes: a case study from the Aegean Bronze Age

Time & Citation Networks

Citation networks emerge from a number of different social systems, such as academia (from publis... more Citation networks emerge from a number of different social systems, such as academia (from published papers), business (through patents) and law (through legal judgements). A citation represents a transfer of information, and so studying the structure of the citation network will help us understand how knowledge is passed on. What distinguishes citation networks from other networks is time; documents can only cite older documents. We propose that existing network measures do not take account of the strong constraint imposed by time. We will illustrate our approach with two types of causally aware analysis. We apply our methods to the citation networks formed by academic papers on the arXiv, to US patents and to US Supreme Court judgements. We show that our tools can reveal that citation networks which appear to have very similar structure by standard network measures, turn out to have significantly different properties. We interpret our results as indicating that many papers in a bi...

The longest path in the Price model

Scientific Reports, 2020

The Price model, the directed version of the Barabási–Albert model, produces a growing directed a... more The Price model, the directed version of the Barabási–Albert model, produces a growing directed acyclic graph. We look at variants of the model in which directed edges are added to the new vertex in one of two ways: using cumulative advantage (preferential attachment) choosing vertices in proportion to their degree, or with random attachment in which vertices are chosen uniformly at random. In such networks, the longest path is well defined and in some cases is known to be a better approximation to geodesics than the shortest path. We define a reverse greedy path and show both analytically and numerically that this scales with the logarithm of the size of the network with a coefficient given by the number of edges added using random attachment. This is a lower bound on the length of the longest path to any given vertex and we show numerically that the longest path also scales with the logarithm of the size of the network but with a larger coefficient that has some weak dependence on...

How the network properties of shareholders vary with investor type and country

PLOS ONE, 2019

Social success of perfumes

PLOS ONE, 2019

A Notation Notation, used throughout this work is summarised in Table 1. Notation Description N T... more

Embedding graphs in Lorentzian spacetime

PloS one, 2017

Geometric approaches to network analysis combine simply defined models with great descriptive pow... more Geometric approaches to network analysis combine simply defined models with great descriptive power. In this work we provide a method for embedding directed acyclic graphs (DAG) into Minkowski spacetime using Multidimensional scaling (MDS). First we generalise the classical MDS algorithm, defined only for metrics with a Riemannian signature, to manifolds of any metric signature. We then use this general method to develop an algorithm which exploits the causal structure of a DAG to assign space and time coordinates in a Minkowski spacetime to each vertex. As in the causal set approach to quantum gravity, causal connections in the discrete graph correspond to timelike separation in the continuous spacetime. The method is demonstrated by calculating embeddings for simple models of causal sets and random DAGs, as well as real citation networks. We find that the citation networks we test yield significantly more accurate embeddings that random DAGs of the same size. Finally we suggest a ...

Was Thebes Necessary? Contingency in Spatial Modeling

Frontiers in Digital Humanities, 2017

Spatial Networks for Archaeology

Randomness and Complexity in Networks

I start by reviewing some basic properties of random graphs. I then consider the role of random w... more I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism. Applying this to networks of fixed size I show that copying and innovation are processes with special mathematical properties which include the ability to solve a simple model exactly for any parameter values and at any time. I finish by looking at variations of this basic model.

Are Copying and Innovation Enough?

Exact analytic solutions and various numerical results for the rewiring of bipartite networks are... more Exact analytic solutions and various numerical results for the rewiring of bipartite networks are discussed. An interpretation in terms of copying and innovation processes make this relevant in a wide variety of physical contexts. These include Urn models and Voter models, and our results are also relevant to some studies of Cultural Transmission, the Minority Game and some models of ecology. Comment: Contribution to the proceedings of ECMI08, based on a talk given by A.D.K.Plato as part of the minisymposium on Mathematics and Social Networks. Includes appendix of supplementary material not in published version

Network rewiring models

Networks & Heterogeneous Media, 2008

Recently we showed that a simple model of network rewiring could be solved exactly for any time a... more Recently we showed that a simple model of network rewiring could be solved exactly for any time and any parameter value. We also showed that this model can be recast in terms of several well known models of statistical physics such as Urn model and the Voter model. We also noted that it has been applied to a wide range of problems. Here we consider various generalisations of this model and include some new exact results.

Exact solution for the time evolution of network rewiring models

Physical Review E, 2007

We consider the rewiring of a bipartite graph using a mixture of random and preferential attachme... more We consider the rewiring of a bipartite graph using a mixture of random and preferential attachment. The full mean field equations for the degree distribution and its generating function are given. The exact solution of these equations for all finite parameter values at any time is found in terms of standard functions. It is demonstrated that these solutions are an excellent fit to numerical simulations of the model. We discuss the relationship between our model and several others in the literature including examples of Urn, Backgammon, and Balls-in-Boxes models, the Watts and Strogatz rewiring problem and some models of zero range processes. Our model is also equivalent to those used in various applications including cultural transmission, family name and gene frequencies, glasses, and wealth distributions. Finally some Voter models and an example of a Minority game also show features described by our model.

A Simple Model of Copying and Innovation

There are many situations where an 'individual' chooses an 'artifacts' by cop... more There are many situations where an 'individual' chooses an 'artifacts' by copying the existing choice of another individual. Names for new babies and registration rates of pedigree dogs often reflect current popular choices [1,2]. The allele for a particular gene carried ('chosen') by an individual reflects current gene frequencies [3]. In Urn models the probabilities controlling the urn chosen by a ball can reflect earlier choices [4]. The individuals in a Voter models [12] copy one of two choices made by a neighbour, as defined by a network of the individuals. Such copying is ...

Identifying time dependence in network growth

Physical Review Research, 2020

Linking the network centrality measures closeness and degree

Communications Physics

Measuring the importance of nodes in a network with a centrality measure is an core task in any n... more Measuring the importance of nodes in a network with a centrality measure is an core task in any network application. There many measures available and it is speculated that many encode similar information. We give an explicit non-linear relationship between two of the most popular measures of node centrality: degree and closeness. Based on a shortest-path tree approximation, we give an analytic derivation that shows the inverse of closeness is linearly dependent on the logarithm of degree. We show that our hypothesis works well for a range of networks produced from stochastic network models and for networks derived from 130 real-world data sets. We connect our results with previous results for other network distance scales such as average distance. Our results imply that measuring closeness is broadly redundant unless our relationship is used to remove the dependence on degree from closeness. The success of our relationship suggests that most networks can be approximated by shortest...

Cycle analysis of Directed Acyclic Graphs

Physica A: Statistical Mechanics and its Applications, 2022

Is network growth time dependent?

Identifying power-law scaling in real networks - indicative of preferential attachment - has prov... more Identifying power-law scaling in real networks - indicative of preferential attachment - has proved controversial. Critics argue that measuring the temporal evolution of a network directly is better than measuring the degree distribution when looking for preferential attachment. However, many of the established methods do not account for any potential time-dependence in the attachment kernels of growing networks, or methods assume that node degree is the key observable determining network evolution. In this paper, we argue that these assumptions may lead to misleading conclusions about the evolution of growing networks. We illustrate this by introducing a simple adaptation of the Barabási-Albert model, the "k2 model", where new nodes attach to nodes in the existing network in proportion to the number of nodes one or two steps from the target node. The k2 model results in time dependent degree distributions and attachment kernels, despite initially appearing to grow as line...

Modelling Cultural Dynamics: A Macroscopic Approach to Cultural Transmission in the Prehistoric Aegean*

INTRODUCTION In this paper we pursue a novel transdisciplinary approach to the question of cultur... more INTRODUCTION In this paper we pursue a novel transdisciplinary approach to the question of cultural dynamics, focussing particularly on the transmission of cultural traits across sociophysical space. In order to think through some of the themes involved we take as a case study an inter-regional phenomenon known as ‘Minoanisation’ (Wiener 1990; Broodbank 2004). This term describes a set of processes observed in the prehistoric Aegean (specifically the Middle and early Late Bronze Age) whereby ‘Minoan’ cultural traits find themselves transmitted beyond the Minoan sphere, i.e. the island of Crete. That is to say, communities in regions such as the Cyclades, the Dodecanese, coastal Asia Minor and the Greek mainland adopt aspects of Cretan culture, both in the form of actual imports from Crete and also local imitations thereof. These cultural elements include pottery shapes and styles, stone vases, loomweights and wall paintings; not only new forms of material culture, but also, presumab...

Time and Citation Networks

ArXiv, 2015

Citation networks emerge from a number of different social systems, such as academia (from publis... more Citation networks emerge from a number of different social systems, such as academia (from published papers), business (through patents) and law (through legal judgements). A citation represents a transfer of information, and so studying the structure of the citation network will help us understand how knowledge is passed on. What distinguishes citation networks from other networks is time; documents can only cite older documents. We propose that existing network measures do not take account of the strong constraint imposed by time. We will illustrate our approach with two types of causally aware analysis. We apply our methods to the citation networks formed by academic papers on the arXiv, to US patents and to US Supreme Court judgements. We show that our tools can reveal that citation networks which appear to have very similar structure by standard network measures turn out to have significantly different properties. We interpret our results as indicating that many papers in a bib...

Articulating scales in sea-based social Networks: A transdisciplinary approach

Dynamic networks and Hamiltonian landscapes: a case study from the Aegean Bronze Age

Time & Citation Networks

Citation networks emerge from a number of different social systems, such as academia (from publis... more Citation networks emerge from a number of different social systems, such as academia (from published papers), business (through patents) and law (through legal judgements). A citation represents a transfer of information, and so studying the structure of the citation network will help us understand how knowledge is passed on. What distinguishes citation networks from other networks is time; documents can only cite older documents. We propose that existing network measures do not take account of the strong constraint imposed by time. We will illustrate our approach with two types of causally aware analysis. We apply our methods to the citation networks formed by academic papers on the arXiv, to US patents and to US Supreme Court judgements. We show that our tools can reveal that citation networks which appear to have very similar structure by standard network measures, turn out to have significantly different properties. We interpret our results as indicating that many papers in a bi...

The longest path in the Price model

Scientific Reports, 2020

The Price model, the directed version of the Barabási–Albert model, produces a growing directed a... more The Price model, the directed version of the Barabási–Albert model, produces a growing directed acyclic graph. We look at variants of the model in which directed edges are added to the new vertex in one of two ways: using cumulative advantage (preferential attachment) choosing vertices in proportion to their degree, or with random attachment in which vertices are chosen uniformly at random. In such networks, the longest path is well defined and in some cases is known to be a better approximation to geodesics than the shortest path. We define a reverse greedy path and show both analytically and numerically that this scales with the logarithm of the size of the network with a coefficient given by the number of edges added using random attachment. This is a lower bound on the length of the longest path to any given vertex and we show numerically that the longest path also scales with the logarithm of the size of the network but with a larger coefficient that has some weak dependence on...

How the network properties of shareholders vary with investor type and country

PLOS ONE, 2019

Social success of perfumes

PLOS ONE, 2019

A Notation Notation, used throughout this work is summarised in Table 1. Notation Description N T... more

Embedding graphs in Lorentzian spacetime

PloS one, 2017

Geometric approaches to network analysis combine simply defined models with great descriptive pow... more Geometric approaches to network analysis combine simply defined models with great descriptive power. In this work we provide a method for embedding directed acyclic graphs (DAG) into Minkowski spacetime using Multidimensional scaling (MDS). First we generalise the classical MDS algorithm, defined only for metrics with a Riemannian signature, to manifolds of any metric signature. We then use this general method to develop an algorithm which exploits the causal structure of a DAG to assign space and time coordinates in a Minkowski spacetime to each vertex. As in the causal set approach to quantum gravity, causal connections in the discrete graph correspond to timelike separation in the continuous spacetime. The method is demonstrated by calculating embeddings for simple models of causal sets and random DAGs, as well as real citation networks. We find that the citation networks we test yield significantly more accurate embeddings that random DAGs of the same size. Finally we suggest a ...

Was Thebes Necessary? Contingency in Spatial Modeling

Frontiers in Digital Humanities, 2017

Spatial Networks for Archaeology

Randomness and Complexity in Networks

I start by reviewing some basic properties of random graphs. I then consider the role of random w... more I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism. Applying this to networks of fixed size I show that copying and innovation are processes with special mathematical properties which include the ability to solve a simple model exactly for any parameter values and at any time. I finish by looking at variations of this basic model.