Why social networks are different from other types of networks

We argue that social networks differ from most other types of networks, including technological and biological networks, in two important ways. First, they have non-trivial clustering or network transitivity, and second, they show positive correlations, also called assortative mixing, between the degrees of adjacent vertices. Social networks are often divided into groups or communities, and it has recently been suggested that this division could account for the observed clustering. We demonstrate that group structure in networks can also account for degree correlations. We show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real-world networks.Comment: 9 pages, 2 figure

Percolation and epidemics in a two-dimensional small world

Percolation on two-dimensional small-world networks has been proposed as a model for the spread of plant diseases. In this paper we give an analytic solution of this model using a combination of generating function methods and high-order series expansion. Our solution gives accurate predictions for quantities such as the position of the percolation threshold and the typical size of disease outbreaks as a function of the density of "shortcuts" in the small-world network. Our results agree with scaling hypotheses and numerical simulations for the same model.Comment: 7 pages, 3 figures, 2 table

Renormalization group analysis of the small-world network model

We study the small-world network model, which mimics the transition between regular-lattice and random-lattice behavior in social networks of increasing size. We contend that the model displays a normal continuous phase transition with a divergent correlation length as the degree of randomness tends to zero. We propose a real-space renormalization group transformation for the model and demonstrate that the transformation is exact in the limit of large system size. We use this result to calculate the exact value of the single critical exponent for the system, and to derive the scaling form for the average number of "degrees of separation" between two nodes on the network as a function of the three independent variables. We confirm our results by extensive numerical simulation.Comment: 4 pages including 3 postscript figure

An analysis of the fixation probability of a mutant on special classes of non-directed graphs

There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. (Lieberman et al. 2005 Nature 433, 312–316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations

Graph Metrics for Temporal Networks

Temporal networks, i.e., networks in which the interactions among a set of elementary units change over time, can be modelled in terms of time-varying graphs, which are time-ordered sequences of graphs over a set of nodes. In such graphs, the concepts of node adjacency and reachability crucially depend on the exact temporal ordering of the links. Consequently, all the concepts and metrics proposed and used for the characterisation of static complex networks have to be redefined or appropriately extended to time-varying graphs, in order to take into account the effects of time ordering on causality. In this chapter we discuss how to represent temporal networks and we review the definitions of walks, paths, connectedness and connected components valid for graphs in which the links fluctuate over time. We then focus on temporal node-node distance, and we discuss how to characterise link persistence and the temporal small-world behaviour in this class of networks. Finally, we discuss the extension of classic centrality measures, including closeness, betweenness and spectral centrality, to the case of time-varying graphs, and we review the work on temporal motifs analysis and the definition of modularity for temporal graphs.Comment: 26 pages, 5 figures, Chapter in Temporal Networks (Petter Holme and Jari Saram\"aki editors). Springer. Berlin, Heidelberg 201

Edge overload breakdown in evolving networks

We investigate growing networks based on Barabasi and Albert's algorithm for generating scale-free networks, but with edges sensitive to overload breakdown. the load is defined through edge betweenness centrality. We focus on the situation where the average number of connections per vertex is, as the number of vertices, linearly increasing in time. After an initial stage of growth, the network undergoes avalanching breakdowns to a fragmented state from which it never recovers. This breakdown is much less violent if the growth is by random rather than preferential attachment (as defines the Barabasi and Albert model). We briefly discuss the case where the average number of connections per vertex is constant. In this case no breakdown avalanches occur. Implications to the growth of real-world communication networks are discussed.Comment: To appear in Phys. Rev.

Analysis of Computer Science Communities Based on DBLP

It is popular nowadays to bring techniques from bibliometrics and scientometrics into the world of digital libraries to analyze the collaboration patterns and explore mechanisms which underlie community development. In this paper we use the DBLP data to investigate the author's scientific career and provide an in-depth exploration of some of the computer science communities. We compare them in terms of productivity, population stability and collaboration trends.Besides we use these features to compare the sets of topranked conferences with their lower ranked counterparts.Comment: 9 pages, 7 figures, 6 table

Citation Networks in High Energy Physics

The citation network constituted by the SPIRES data base is investigated empirically. The probability that a given paper in the SPIRES data base has $k$ citations is well described by simple power laws, $P(k) \propto k^{-\alpha}$, with $\alpha \approx 1.2$ for $k$ less than 50 citations and $\alpha \approx 2.3$ for 50 or more citations. Two models are presented that both represent the data well, one which generates power laws and one which generates a stretched exponential. It is not possible to discriminate between these models on the present empirical basis. A consideration of citation distribution by subfield shows that the citation patterns of high energy physics form a remarkably homogeneous network. Further, we utilize the knowledge of the citation distributions to demonstrate the extreme improbability that the citation records of selected individuals and institutions have been obtained by a random draw on the resulting distribution.Comment: 9 pages, 6 figures, 2 table

Cross-over behaviour in a communication network

We address the problem of message transfer in a communication network. The network consists of nodes and links, with the nodes lying on a two dimensional lattice. Each node has connections with its nearest neighbours, whereas some special nodes, which are designated as hubs, have connections to all the sites within a certain area of influence. The degree distribution for this network is bimodal in nature and has finite variance. The distribution of travel times between two sites situated at a fixed distance on this lattice shows fat fractal behaviour as a function of hub-density. If extra assortative connections are now introduced between the hubs so that each hub is connected to two or three other hubs, the distribution crosses over to power-law behaviour. Cross-over behaviour is also seen if end-to-end short cuts are introduced between hubs whose areas of influence overlap, but this is much milder in nature. In yet another information transmission process, namely, the spread of infection on the network with assortative connections, we again observed cross-over behaviour of another type, viz. from one power-law to another for the threshold values of disease transmission probability. Our results are relevant for the understanding of the role of network topology in information spread processes.Comment: 12 figure

Scaling Properties of Random Walks on Small-World Networks

Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in $d$-dimensions. At intermediate times, the properties of the walker shows scaling behavior characteristic of an infinite small-world network. Finally, at long times, the finite size of the network becomes important, and many of the properties of the walker saturate. We propose general analytical forms for the scaling properties in all three regimes, and show that these analytical forms are consistent with our numerical simulations.Comment: 7 pages, 8 figures, two-column format. Submitted to PR