Regarding the azimuthally integrated, radial distribution of bright lumps - corresponding to star-forming complexes - in dwarf irregular galaxies, we find them non-uniformly distributed. While in individual galaxies the number distribution is non-monotonic and rugged, the summed-up distribution for all galaxies of our sample manifests the hidden constraint, which is a r e^-r-distribution closely tracing the underlying older population. More precisely, in terms of radial number density distribution the lumps follow an exponential decay with scale length about 10 percent smaller on average than that of the blue continuum light. This is consistent with studies of the radial distribution of H II regions in intermediate-type spiral galaxies (Athanassoula et al. 1998). The fact that each component of the average disk - from star-forming site number density to surface brightness of the total light - is approximately exponential is a hint that luminous exponential disks are born rather than made, consistent with the accretion scenario for the viscous evolution of galaxy disks (e.g., Ferguson & Clarke 2001). Star-forming complexes in irregular dwarf galaxies can be found out to large radii, as already emphasized by Schulte-Ladbeck & Hopp (1998), Brosch et al. (1998), and Roye & Hunter (2000). The presence of a tail in the accumulated radial number distribution of star forming regions out to at least six optical scale lengths (Fig. 4) indicates that the distributions of dwarf irregulars are truncated at rather low gas density thresholds for star formation. (van Zee et al. 1997; Hunter et al. 1998; Pisano et al. 2000); this seems to be different with many spiral galaxies for which sharp to weak truncations, starting at galactocentric radii of 2-4 near-infrared or 3-5 optical scale lengths, have persistently been reported (e.g., recently, Florido et al. 2001; Kregel et al. 2002; Pohlen et al. 2002).

Beside the presence of main or primary peaks in the radial lump number distribution at slightly less than one optical scale length on average, there is - contrary to the expectations for exponential-disk systems - the frequent occurence of secondary peaks at about two scale lengths. As simple simulations show, it is consistent with the idea of triggered star formation based on a stochastic self-regulation scenario. However, some of the brighter, larger galaxies exhibit pronounced primary peaks at two scale lengths and show minor, secondary peaks around one scale length. For these galaxies with a reversed peak pattern the simulations indicate that shear-induced star formation around the disk's turnover to differential rotation could be at work; we feel this issue worth a deeper investigation: given a lump statistics based on higher-resolution images and linked to detailed rotational velocity data, and possibly supplemented with information on large-scale magnetic field structures, this may lead to some subtle but decisive insights related to star formation in irregular galaxies. Also, the possible role of bars or bar-like central features should be carefully considered. However, as none of our four galaxies with principal peaks appering around two scale lengths is classified as "barred'', and because Roye & Hunter (2000) did not see a preferential location of H $\;$ II regions towards the ends of bars in the two candidate galaxies of their sample, we do not consider this mechanism as being effective in shaping the number distribution of lumps.

The observation that the scale lengths are the larger the older the underlying respective population is, goes in hand with the finding that in star-forming dwarf galaxies the oldest populations are also the most extended ones (Gardiner & Hatzidimitriou 1992; Minniti & Zijlstra 1996; Minniti et al. 1999; Harris & Zaritsky 1999). The common interpretation is that of an age-related dispersion of stars. Because dynamical disk heating tends to saturate at a fixed velocity dispersion (Freeman & Bland-Hawthorn 2002) the amount of radial spread introduced by dynamical heating is expected to be larger in smaller galaxies with lower gravitational binding energies (J. Gallagher, private communication). Indeed, the fact noted in Parodi et al. (2002) that with increasing scale length the disk color gradients become systematically less positive and even start being weakly negative when going from dwarf irregular to low-surface brightness and spiral galaxies - which is equivalent to larger galaxies having red-to-blue band scale length ratios below one - seems to support the idea that the extent of the red component decreases as galaxy mass increases. However, for spirals part of this colour trend is most certainly a metallicity effect.

We provide no investigation of the azimuthal lump distribution; where necessary we assumed axisymmetry. Applying different asymmetry indices to dwarf and normal irregulars, Heller et al. (2000) and Roye & Hunter (2000) obtained opposing results that were dependent on the definition of the chosen index (asymmetry as a question of perception). Also, it was found that while the conveniently applied rotational asymmetry index may be used as a first discriminator between (distant) elliptical and spiral/irregular galaxies (Schade et al. 1995; Conselice et al. 2000), this parameter actually is strongly dependent on recent star formation (Takamiya 1999; Mayya & Romano 2001) and thus must be correlated with the lumpiness index as adopted in the present paper. As we have no data for the star formation rate of our galaxies, an immediate comparision cannot be offered, however.

Applying a concentration index CI that is normalized according to the exponential-disk structure of a mean lump distribution leads to consistent results for varying aperture sizes. It may also remove the discrepancy found by Heller et al. (2000) between the $CI_{\rm H\alpha}$ values for actual galaxies and the lower ones for simulated galaxies with random star formation region positions. Roye & Hunter (2000) pointed out an increased scatter of concentration indices for faster rotating galaxies of their sample; we no longer see this effect with our larger sample.

Comparing concentration (CI) with lumpiness ( $\chi$ ) we find the galaxies with a high percentage (> $10\%$ ) of light stemming from the lumps showing low to moderate concentrations ( $CI \la 2$ ), i.e., galaxies with lumps that are widely scatterd within the disk maintain a higher fraction of the total Bluminosity (Fig. 8). On the other hand, for very concentrated galaxies ( $CI \ga 2$ ) less than about 10 percent of the light is due to the lumps; actually, for these galaxies the values for the lumpiness index $\chi$ fluctuate around the sharp value of 7% attributed to most of the (barred) spiral and irregular galaxies observed by Elmegreen & Salzer (1999). They suggested the similarity of the blue-band light fraction in complexes for several galaxies of different Hubble types and different total luminosities being due to similar star formation efficiencies.

It remains, however, remarkable that galaxies with very high lump concentrations are not among the galaxies showing high Bluminosity fractions of the lumps (Fig. 8, bottom panel). One may wonder whether some of the nearby, well-known BCDs with central starbursts would agree with this conclusion as well, i.e. whether their central bursts should not dominate the total light content. We therefore examined where the typical starburst dwarf irregular galaxies NGC 1569 and NGC 1705 would fill in the CI vs. $\chi$ diagram. Based on the 48 brightest central star clusters of NGC 1569 as compiled in Hunter et al. (2000) and adopting a distance of 2.5 Mpc and a galaxy absolute magnitude of M_V=-17.99mag, we estimate a mere $\chi \le 0.10$ for NGC 1569. Similarly, for the super-star cluster dominated amorphous galaxy NGC 1705, the absolute magnitude for the 7 brightest clusters is about M_V=-14.1according to the data in O'Connell et al. (1994), whereas the galaxy has a magnitude of M_V=-16.13 at a distance of 5 Mpc, implying only $\chi \approx 0.15$ . Thus independent of the exact concentration indices for the brightest clusters, these two small but representative galaxies with central starbursts would not occupy the empty part of the CI vs. $\chi$ diagram. This means that the empty region in this figure is not a selection effect, but may be related to our adopted procedures in determining the corresponding indices.

While the concentration index is a measure for lump spreading, the cluster or correlation dimension provides information on the scaling behaviour for lump-to-lump distances. We found the lump cluster dimensions - corrected for the effect of radial abundances according to the annulus-integrated exponential distribution - to lie between 1.3 and 2.0 and to gently correlate with extrapolated central surface brightness and absolute magnitude of the host galaxy. At the same time the cluster dimension is weakly anticorrelated with the lumpiness index.

$\begin{figure} \par\includegraphics[width=18cm,clip]{MS3024f09.ps}\end{figure}$

Figure 9: SSPSF simulation of an exponentiated disk with 80 corotating rings and a transition from rigid to strong differential rotation at two scale lengths (ring 32, solid line), after 500 time steps. The three panels show the pattern of newly activated cells (filled circles) together with those created less than 10 time steps ago (dots), the radial number distribution of the active cells (bin width is 8 rings or half a scale length), and the corresponding number density distribution. Note that for the number distribution the peak at one scale length (ring 16) - the presence of which is expected for non-differentially rotating exponential disks - is markedly surpassed by a peak around two scale lengths.

Cluster dimension (or porosity) as introduced in this paper may be intimately linked to the sizes of the largest, kiloparsec-sized lump compounds (Elmegreen et al. 1996) or to the sizes of star-forming, collapsed expanding shells (e.g., Walter 1999; Valdez-Gutiérrez et al. 2001). Elmegreen et al. (1996) found the sizes of the largest compounds within spiral and irregular galaxies to approximately scale with the square root of the galaxy luminosity, or, if normalized by the galaxy semi-major axis R₂₅, small galaxies to have slightly larger relative compound diameters than larger galaxies. This was hypothesized to result from gravitational instabilities with the Jeans length or the mean virial density varying with galaxy luminosity. Walter (1999) suggests the holes in dwarf irregulars to be larger than those in late-type spirals because small galaxies have lower masses and correspondingly lower gravitational potentials and lower ambient ISM gas densities, favoring H I shells to grow larger. Alternatively, we are tentatively interpreting the cluster dimension of a galaxy in terms of the volume filling factor of empty regions in a fractal medium, i.e. in terms of porosity (defined analogous to Elmegreen 1997), and find the following statistical trends: (i) fainter galaxies tend to be more porous; (ii) more porous galaxies also have a lumpier morphology with lower central lump concentrations; (iii) the more porous the galaxy, the lower the star formation rate per kpc² (Eq. (4)). While these trends are not unexpected, we provide an objective and quantitative statistical treatment of these.

Porosity, or self-similarity, as observed with the bright-lump distribution within dwarf irregular galaxies reflects the self-regulated evolution of the interstellar medium, with stellar feedback and self-gravitation being the main mechanisms. It is thus not to be confused with the still self-similar pattern of dispersed stellar aggregates that initially formed from the fractal interstellar gas, obeying the canonical value of D=1.3 for turbulence-driven star formation (Elmegreen & Elmegreen 2001). Our sub-galactic areas are showing $D \ga 1.7$ , i.e. much higher fractal dimensions. We suspect that in dwarf irregular galaxies either the dispersive redistribution of stars is indeed much more effective than in spiral galaxies (Elmegreen & Hunter 2000), and/or that feed-back regulation is responsible for the partially randomized position patterns (e.g., due to the intersection of giant shells).

7 Discussion and conclusions