Mikko Kivelä

Small But Slow World: How Network Topology and Burstiness Slow Down Spreading

While communication networks show the small-world property of short paths, the spreading dynamics in them turns out slow. Here, the time evolution of information propagation is followed through communication networks by using empirical data on contact sequences and the susceptible-infected model. Introducing null models where event sequences are appropriately shuffled, we are able to distinguish between the contributions of different impeding effects. The slowing down of spreading is found to be caused mainly by weight-topology correlations and the bursty activity patterns of individuals.

Mathematical Formulation of Multilayer Networks

A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems are very rich. Achieving a deep understanding of such systems necessitates generalizing ‘‘traditional’’ network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multilayer complex systems. In this paper, we introduce a tensorial framework to study multilayer networks, and we discuss the generalization of several important network descriptors and dynamical processes—including degree centrality, clustering coefficients, eigenvector centrality, modularity, von Neumann entropy, and diffusion—for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multilayer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems.

Generalizations of the clustering coefficient to weighted complex networks.

The recent high level of interest in weighted complex networks gives rise to a need to develop new measures and to generalize existing ones to take the weights of links into account. Here we focus on various generalizations of the clustering coefficient, which is one of the central characteristics in the complex network theory. We present a comparative study of the several suggestions introduced in the literature, and point out their advantages and limitations. The concepts are illustrated by simple examples as well as by empirical data of the world trade and weighted coauthorship networks.

Sequential algorithm for fast clique percolation

In complex network research clique percolation, introduced by Palla, Derényi, and Vicsek [Nature (London) 435, 814 (2005)], is a deterministic community detection method which allows for overlapping communities and is purely based on local topological properties of a network. Here we present a sequential clique percolation algorithm (SCP) to do fast community detection in weighted and unweighted networks, for cliques of a chosen size. This method is based on sequentially inserting the constituent links to the network and simultaneously keeping track of the emerging community structure. Unlike existing algorithms, the SCP method allows for detecting k -clique communities at multiple weight thresholds in a single run, and can simultaneously produce a dendrogram representation of hierarchical community structure. In sparse weighted networks, the SCP algorithm can also be used for implementing the weighted clique percolation method recently introduced by Farkas [New J. Phys. 9, 180 (2007)]. The computational time of the SCP algorithm scales linearly with the number of k -cliques in the network. As an example, the method is applied to a product association network, revealing its nested community structure.

A comparative study of social network models: Network evolution models and nodal attribute models

This paper reviews, classifies and compares recent models for social networks that have mainly been published within the physics-oriented complex networks literature. The models fall into two categories: those in which the addition of new links is dependent on the (typically local) network structure (network evolution models, NEMs), and those in which links are generated based only on nodal attributes (nodal attribute models, NAMs). An exponential random graph model (ERGM) with structural dependencies is included for comparison. We fit models from each of these categories to two empirical acquaintance networks with respect to basic network properties. We compare higher order structures in the resulting networks with those in the data, with the aim of determining which models produce the most realistic network structure with respect to degree distributions, assortativity, clustering spectra, geodesic path distributions, and community structure (subgroups with dense internal connections). We find that the nodal attribute models successfully produce assortative networks and very clear community structure. However, they generate unrealistic clustering spectra and peaked degree distributions that do not match empirical data on large social networks. On the other hand, many of the network evolution models produce degree distributions and clustering spectra that agree more closely with data. They also generate assortative networks and community structure, although often not to the same extent as in the data. The ERGM model, which turned out to be near-degenerate in the parameter region best fitting our data, produces the weakest community structure.

Multiscale analysis of spreading in a large communication network

In temporal networks, both the topology of the underlying network and the timings of interaction events can be crucial in determining how a dynamic process mediated by the network unfolds. We have explored the limiting case of the speed of spreading in the SI model, set up such that an event between an infectious and a susceptible individual always transmits the infection. The speed of this process sets an upper bound for the speed of any dynamic process that is mediated through the interaction events of the network. With the help of temporal networks derived from large-scale time-stamped data on mobile phone calls, we extend earlier results that indicate the slowing-down effects of burstiness and temporal inhomogeneities. In such networks, links are not permanently active, but dynamic processes are mediated by recurrent events taking place on the links at specific points in time. We perform a multiscale analysis and pinpoint the importance of the timings of event sequences on individual links, their correlations with neighboring sequences, and the temporal pathways taken by the network-scale spreading process. This is achieved by studying empirically and analytically different characteristic relay times of links, relevant to the respective scales, and a set of temporal reference models that allow for removing selected time-domain correlations one by one. Our analysis shows that for the spreading velocity, time-domain inhomogeneities are as important as the network topology, which indicates the need to take time-domain information into account when studying spreading dynamics. In particular, results for the different characteristic relay times underline the importance of the burstiness of individual links.

Using explosive percolation in analysis of real-world networks.

We apply a variant of the explosive percolation procedure to large real-world networks and show with finite-size scaling that the university class, ordinary or explosive, of the resulting percolation transition depends on the structural properties of the network, as well as the number of unoccupied links considered for comparison in our procedure. We observe that in our social networks, the percolation clusters close to the critical point are related to the community structure. This relationship is further highlighted by applying the procedure to model networks with predefined communities.

Networks of Emotion Concepts

The aim of this work was to study the similarity network and hierarchical clustering of Finnish emotion concepts. Native speakers of Finnish evaluated similarity between the 50 most frequently used Finnish words describing emotional experiences. We hypothesized that methods developed within network theory, such as identifying clusters and specific local network structures, can reveal structures that would be difficult to discover using traditional methods such as multidimensional scaling (MDS) and ordinary cluster analysis. The concepts divided into three main clusters, which can be described as negative, positive, and surprise. Negative and positive clusters divided further into meaningful sub-clusters, corresponding to those found in previous studies. Importantly, this method allowed the same concept to be a member in more than one cluster. Our results suggest that studying particular network structures that do not fit into a low-dimensional description can shed additional light on why subjects evaluate certain concepts as similar. To encourage the use of network methods in analyzing similarity data, we provide the analysis software for free use (http://www.becs.tkk.fi/similaritynets/).

Estimating interevent time distributions from finite observation periods in communication networks.

A diverse variety of processes-including recurrent disease episodes, neuron firing, and communication patterns among humans-can be described using interevent time (IET) distributions. Many such processes are ongoing, although event sequences are only available during a finite observation window. Because the observation time window is more likely to begin or end during long IETs than during short ones, the analysis of such data is susceptible to a bias induced by the finite observation period. In this paper, we illustrate how this length bias is born and how it can be corrected without assuming any particular shape for the IET distribution. To do this, we model event sequences using stationary renewal processes, and we formulate simple heuristics for determining the severity of the bias. To illustrate our results, we focus on the example of empirical communication networks, which are temporal networks that are constructed from communication events. The IET distributions of such systems guide efforts to build models of human behavior, and the variance of IETs is very important for estimating the spreading rate of information in networks of temporal interactions. We analyze several well-known data sets from the literature, and we find that the resulting bias can lead to systematic underestimates of the variance in the IET distributions and that correcting for the bias can lead to qualitatively different results for the tails of the IET distributions.

Dynamics of Investor Spanning Trees Around Dot-Com Bubble

We identify temporal investor networks for Nokia stock by constructing networks from correlations between investor-specific net-volumes and analyze changes in the networks around dot-com bubble. The analysis is conducted separately for households, financial, and non-financial institutions. Our results indicate that spanning tree measures for households reflected the boom and crisis: the maximum spanning tree measures had a clear upward tendency in the bull markets when the bubble was building up, and, even more importantly, the minimum spanning tree measures pre-reacted the burst of the bubble. At the same time, we find less clear reactions in the minimal and maximal spanning trees of non-financial and financial institutions around the bubble, which suggests that household investors can have a greater herding tendency around bubbles.